A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. Compare with the Kepler-Poinsot solids, which are not convex, and the Archimedean and Johnson solids, which while made of regular polygons are not themselves regular.

There are five Platonic solids, all known to the ancient Greeks:

Limited number of Platonic polyhedra

That there are only five such three-dimensional solids is easily demonstrated. To create a vertex, at least three faces must meet at a point and the total of their angles must be less than 360°, i.e the corners of the face must be less than 360°/3=120°. The only polygons meeting these requirements are the triangle, square, and pentagon.

• Triangular faces: each vertex of a regular triangle is 60°, so a shape should be possible with 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
• Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
• Pentagonal faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron, and that exhausts the list of regular 3-dimensional solids.

Dual polyhedra

Note that if you connect the centers of the faces of a tetrahedron, you get another tetrahedron. If you connect the centers of the faces of an octahedron, you get a cube, and vice versa. If you connect the centers of the faces of a dodecahedron, you get an icosahedron, and vice versa. These pairs are said to be dual polyhedra.

Origins of name

The Platonic solids are named after Plato, who wrote about them in Timaeus. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of Euclid's Elements. Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.

Ancient symbolism

Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids that had been discovered by the Pythagoreans (in the Timaeus). These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium fluoride) in octahedra, and pyrite in dodecahedra (see uses below).

This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. There was logical reasoning behind these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and breaks when picked up, in stark difference to the smooth flow of water.

The fifth Platonic Solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven" (Timaeus 55). He didn't really know what else to do with it. Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

Other symbolism

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements.

Inscribed Platonic polyhedra

When the Platonic polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:

• Tetrahedron: 12.2518%
• Cube: 36.7553%
• Octahedron: 31.8310%
• Dodecahedron: 66.4909%
• Icosahedron: 60.5461%

The Platonic solids may be seen as increasingly better approximations to that sphere. (The Archimedean solids and geodesic domes are in many ways even better approximations to the sphere).

However, either the dodecahedron or the icosahedron may be seen as the Platonic solid that "best approximates" a sphere.

On one hand, the icosahedron has the most sides and the flattest dihedral angle. This may be the source of the common assumption that the the icosahedron is the Platonic solid that gives the closest approximation to the sphere.

On the other hand, the dodecahedron occupies significantly more of the sphere's volume than the apparently more spherical icosahedron. The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere.

The dodecahedron is also most like the sphere in the sense that it has the smallest central angle (ratio of the strut length to the radius of the circumscribed sphere), and the greatest surface area. [1] (http://kjmaclean.com/Geometry/Platonic.html)

In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron. Both have an inscribed sphere whose radius is (φ+1) / ( 2 * √( 3 * φ + 6 ) ) (about 0.795) of the radius of the outer sphere.

Uses

The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.).

The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube.

External links

• Stella: Polyhedron Navigator (http://www.software3d.com/Stella.html) Tool for exploring polyhedra
• Paper Models of Polyhedra (http://www.korthalsaltes.com/) Many links
• The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
• Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra
• London South Bank University (http://www.lsbu.ac.uk/water/platonic.html) Water structure and behavior
• Book XIII (http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html) of Euclid's Elements.